dHSIC (version 1.0)

dhsic: d-variable Hilbert Schmidt independence criterion - dHSIC

Description

The d-variable Hilbert Schmidt independence criterion (dHSIC) is a non-parametric measure of dependence between an arbitrary number of variables. In the large sample limit the value of dHSIC is 0 if the variables are jointly independent and positive if there is a dependence. It is therefore able to detect any type of dependence given a sufficient amount of data.

Usage

dhsic(X, Y, kernel = "gaussian")

Arguments

X

either a list of at least two numeric matrices or a single numeric matrix. The rows of a matrix correspond to the observations of a variable. It is always required that there are an equal number of observations for all variables (i.e. all matrices have to have the same number of rows). If X is a single numeric matrix than one has to specify the second variable as Y.

Y

a numeric matrix if X is also a numeric matrix and omitted if X is a list.

kernel

a vector of character strings specifying the kernels for each variable. There exist two pre-defined kernels: "gaussian" (Gaussian kernel with median heuristic as bandwidth) and "discrete" (discrete kernel). User defined kernels can also be used by passing the function name as a string, which will then be matched using match.fun. If the length of kernel is smaller than the number of variables the kernel specified in kernel[1] will be used for all variables.

Value

A list containing the following components:

dHSIC

the value of the empirical estimator of dHSIC

time

numeric vector containing computation times. time[1] is time to compute Gram matrix and time[2] is time to compute dHSIC.

Details

The d-variable Hilbert Schmidt independence criterion is a direct extension of the standard Hilbert Schmidt independence criterion (HSIC) from two variables to an arbitrary number of variables. It is 0 if and only if all the variables are jointly independent. This function computes an estimator of dHSIC, which converges to the actual dHSIC in the large sample limit. It is therefore possible to detect any type of dependence in the large sample limit. For more details see the references.

References

Gretton, A., K. Fukumizu, C. H. Teo, L. Song, B. Sch\"olkopf and A. J. Smola (2007). A kernel statistical test of independence. In Advances in Neural Information Processing Systems (pp. 585-592). Pfister, N., P. B\"uhlmann, B. Sch\"olkopf and J. Peters (2016). Kernel-based Tests for Joint Independence. ArXiv e-prints (1603.00285).

See Also

dhsic.test

Examples

### Two different input methods
set.seed(0)
x <- matrix(rnorm(200),ncol=2)
y <- matrix(rbinom(100,30,0.1),ncol=1)
dhsic(list(x,y),kernel=c("gaussian","discrete"))$dHSIC
dhsic(x,y,kernel=c("gaussian","discrete"))$dHSIC

### Using a user-defined kernel (here: sigmoid kernel)
set.seed(0)
x <- matrix(rnorm(500),ncol=1)
y <- x^2+0.02*matrix(rnorm(500),ncol=1)
sigmoid <- function(x_1,x_2){
  return(tanh(sum(x_1*x_2)))
}
dhsic(x,y,kernel="sigmoid")$dHSIC